Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 79 / 3

Define the positive total shear stress by $\tau =\rho [\nu U_z-\overline{u^{\prime }{w}^{\prime }}]$, so the physical wall traction on the fluid opposes the positive flow. The steady mean momentum equation givesFor a driving pressure gradient $P_x<0$, the stress decreases upward. It is approximately constant whileAn outer boundary-layer height can impose a stricter limit. In the turbulent region outside the viscous or roughness sublayer, the only local neutral scales are $u_{\ast }$ and $z$, so $U_z=C{u}_{\ast }/z$. Conventionally write $C=1/\kappa$, where $\kappa$ is the von Kármán constant. Equivalently, a mixing length $\ell =\kappa z$ and stress closure $u_{\ast }^2={\ell }^2{U}_z^2$ give the same result. Integration yields the law of the wallThe roughness length $z_0$ is an extrapolation scale, not a statement that the logarithm is valid at the actual solid surface. For a smooth boundary, the viscous inner scales are $z^{+}=z{u}_{\ast }/\nu$, $U^{+}=U/u_{\ast }$. The viscous sublayer has $U^{+}\simeq z^{+}$; farther out $U^{+}={\kappa }^{-1}\log z^{+}+C_s$, corresponding to $z_0=(\nu /u_{\ast })e^{-\kappa C_s}$. For a fully rough boundary, $z_0$ is proportional to a geometric roughness height and viscosity no longer determines the leading profile. The logarithmic region still needs roughness height $\ll z\ll {\delta }_{\tau }$.

In local equilibrium, production of turbulent kinetic energy by the shear is $P=-\overline{u^{\prime }{w}^{\prime }}{U}_z\simeq u_{\ast }^2{U}_z$. Neglecting its transport relative to production and viscous dissipation,Dimensional similarity alone gives $ϵ\propto u_{\ast }^3/z$; the displayed coefficient additionally uses the local energy balance and the logarithmic shear.

Let $B_0=\overline{w^{\prime }{b}^{\prime }}$ be the vertical turbulent buoyancy flux, positive for heating from below. Its dimensions are length squared per time cubed. The stress velocity and height therefore admit a new dimensionless group $B_{0}z/u_{\ast }^3$. Define the signed Obukhov lengthStable cooling has $B_0<0,{\ell }_O>0$; unstable heating has $B_0>0,{\ell }_O<0$. Monin-Obukhov similarity theory expresses the mean shear asThe similarity hypothesis assumes approximately steady, horizontally homogeneous, constant-stress and constant-flux conditions, with the observation level outside the inner sublayer but below the outer-layer scales. Dimensional analysis fixes the arguments and prefactor, not the entire function ${\varphi }_m$. A measured or modeled turbulence closure supplies its form.

For a rough boundary, integrate the similarity shear from the roughness length. A usual moderate-stability closure iswith positive dimensionless coefficients. These are empirical closures, not quantities determined by similarity alone. An often-used choice is $c_s=5,c_u=16$. The unstable quarter-power relation is tested over approximately $-1\lesssim z/{\ell }_O\lesssim -0.01$ in Dyer's flux-gradient study; the stable coefficient convention is recorded in the surface-flux formulation. Near neutrality one takes their smooth limits.

For moderate stable conditions, integration givesFor the unstable closure set $x(\zeta )=(1-c_u\zeta {)}^{1/4}$ and defineThen $1-\zeta {\psi }_m^{\prime }(\zeta )={\varphi }_m(\zeta )$ andFor weak unstable departure, $|c_{u}z/{\ell }_O|\ll 1$, this reduces to the stable-form expression with $c_s$ replaced by $c_u/4$ and a negative ${\ell }_O$. The full quarter-power expression is useful for moderately unstable conditions with $|z/{\ell }_O|$ of order one; it must not be extrapolated automatically to the distinct asymptotic free-convection regime. The stable linear expression similarly describes a continuously turbulent surface layer, rather than guaranteeing validity after strong stratification destroys the constant-flux state. Both profiles require $z_0\ll z\ll {\delta }_{\tau }$ and approximately uniform fluxes.

At fixed friction velocity, unstable heating gives $B_0>0$ and $\zeta <0$. The usual similarity closure has ${\varphi }_m<1$, so it reduces the mean velocity gradient relative to the neutral law of the wall. Rising warm and descending cool parcels mix momentum efficiently; less mean shear is needed to transport the same stress. Relative to a fixed near-wall reference, the speed difference aloft is reduced.

Stable cooling gives $B_0<0,\zeta >0$ and ${\varphi }_m>1$. The restoring buoyancy suppresses vertical parcel excursions and reduces the effective momentum diffusivity, requiring a steeper shear for the same stress. The speed difference from the wall reference is correspondingly larger. These are comparisons at prescribed stress; fixing an outer wind instead would allow $u_{\ast }$ and the stress to change, so it would not be the same comparison.

Write $S=U_z$ and $N^2=d\bar{b}/dz$. The gradient Richardson number and Flux Richardson number areThe first compares the restoring buoyancy gradient with mean shear; the second is the fraction of shear production consumed by stable buoyancy work. With this convention, both are positive for stable mixing and negative for unstable stratification. In local steady equilibrium, $ϵ=P+B_0=P(1-{\mathrm{Ri}}_f)$, so a positive dissipation requires ${\mathrm{Ri}}_f<1$. The inviscid Miles–Howard theorem concerning ${\mathrm{Ri}}_g\ge 1/4$ is a linear stability criterion, not a universal equality for turbulent fluxes.

Use the gradient-diffusion closures $-\overline{u^{\prime }{w}^{\prime }}=K_{m}S$ and $B_0=-K_b{N}^2$. The turbulent Prandtl number is ${\mathrm{Pr}}_t=K_m/K_b$, givingUnder the ideal Reynolds analogy $K_m=K_b$, the two Richardson numbers coincide. It is an additional closure, not an automatic consequence of the definitions. If the buoyancy-gradient similarity function is defined by $N^2=-B_0{\varphi }_h/(\kappa z{u}_{\ast })$, then

For lower-boundary heating, forced convection means that mean shear supplies most of the turbulence even though buoyancy assists it, approximately $z\ll |{\ell }_O|$. Free convection means buoyancy dominates, approximately $z\gg |{\ell }_O|$ within the constant-flux region. In this limit the local convective speed isIt is the velocity whose eddy transport supplies the imposed vertical turbulent buoyancy flux. The associated diffusivity is $K_b=C_K{w}_{c}z=C_K{B}_0^{1/3}{z}^{4/3}$, with a dimensionless coefficient fixed by a closure. From $B_0=-K_b{N}^2$,Here $N^2<0$: there is no real restoring buoyancy frequency. The displayed magnitude is a local overturning/growth rate, or one may write $N=i|N|$. The Reynolds analogy makes $K_m=K_b$ and hence $U_z\sim u_{\ast }^2{B}_0^{-1/3}{z}^{-4/3}$; the remaining mean shear is weak compared with the convective motion.

The turnover dissipation estimate givesalso consistent with the local energy equation $ϵ=P+B_0$ when $P$ is negligible. Thus free-convection dissipation is height independent to leading scaling order, in contrast with neutral $u_{\ast }^3/(\kappa z)$. The constants and detailed profiles are not determined by dimensional analysis. These are local free-convection surface-layer scalings, not a claim that an entire deep convecting layer has a uniform unstable mean buoyancy gradient.

For very stable continuously turbulent conditions, $z\gg {\ell }_O>0$, local eddies lose leading dependence on their distance from the wall. In very stable surface-layer similarity, the dimensionless functions must therefore scale as ${\varphi }_m\sim c_m\zeta$ and ${\varphi }_h\sim c_h\zeta$, with positive closure constants. Using ${\ell }_O=u_{\ast }^3/(\kappa |B_0|)$ givesThus the mean velocity is approximately linear in height and the real buoyancy frequency is approximately constant, $N\sim \sqrt{c_h}|B_0|/u_{\ast }^2$. Under the Reynolds analogy, $c_h=c_m$.

The local turbulent kinetic energy balance givesPositive dissipation requires $c_m>1$. The turbulent energy-turnover length is of order $u_{\ast }^3/|B_0|$, hence of order the Obukhov length, independent of observation height. A related buoyancy-limited length is $u_{\ast }/N$. The Ozmidov length follows by equating an eddy turnover time ${\ell }^{2/3}{ϵ}^{-1/3}$ with $N^{-1}$:These are turbulent turnover or buoyancy-transition scales. If “the scale at which dissipation occurs” means the actual viscous cutoff, molecular viscosity is also required: the Kolmogorov microscales giveWithout $\nu$, the viscous length cannot be fixed by the supplied stress and flux alone. Separating it from the turbulent turnover length avoids confusing an energy-budget estimate with the molecular cutoff.

Finally,They become height-independent constants. The commonly used continuation $c_m=c_h=5$ gives ${\mathrm{Ri}}_f={\mathrm{Ri}}_g\simeq 0.2$ and $ϵ\simeq 4|B_0|$. This numerical value is a closure choice, not a deduction from units; setting $c_m=1$ would incorrectly leave zero dissipation. Extremely stable intermittent or collapsed turbulence need not satisfy this local-equilibrium model. The relation between the Obukhov and buoyancy-dissipation lengths in the local regime is also analyzed in Grachev and colleagues' similarity study.